The Inferential-Expressive Trade-Off: A Case Study of Tabular Representations

Many graphical systems (e.g., Euler diagrams, maps, pictorial images, and even tables) support efficient inferences or rich presentation of information apparently at the expense of expressive flexibility. This association of inferential efficiency, expressive richness, and expressive inflexibility in a graphical system has been pointed out by various researchers (e.g., Sloman [1], Stenning and Oberlander [2]). This paper investigates the semantic mechanism of the association by closely examining a particular system of tabular representations, which, despite its simplicity, clearly exhibits all those opposing functional traits. Using a semantic framework of channel theory (Barwise and Seligman [3]), we will show that the common mechanism is a parallelism between abstraction relations in represented properties and in representing properties. Intuitions tell us that both positive and negative traits coexist in graphics. Graphics are rich in content; they facilitate our inferences on the depicted objects or situations; but they often have severe limitations in what can be expressed. Recent studies have started revealing the semantic mechanisms behind these functional traits of graphics individually: their expressive richness (e.g., Kosslyn [4]), their potentials for efficient inferences (e.g., Barwise and Etchemendy [5]), and their expressive inflexibility (e.g., Stenning and Oberlander [2]). Now, is there any relationship among these coexisting traits? Or are they all independent from each other? Many researchers have suggested their connection. Sloman [1] mentions a tradeoff between efficient “problem-solving power” and expressive generality, suggesting that “analogical” representation systems (such as graphical systems) sacrifice the latter for the former (pp. 217). Barwise and Etchemendy [5] mention a trade-off between expressive richness and expressive flexibility. Comparing pictorial images and first-order sentences, they point out that graphics are good at expressing conjunctive information but not at expressing disjunctive information (p. 22). Barwise and Hammer [6] state that the expressive generality of a system is often incompatible with its capacity for being homomorphic to its target domain, while this capacity is a root of a system’s expressive richness (pp. 47–48). It is, however, Stenning and Oberlander [2] who have offered the clearest insight into the connection between positive and negative functional traits of graphics. According to them, “graphical representation such as diagrams limit abstraction and thereby aid processibility” (p. 98). Here, “limiting abstraction” means limited capacities of expressing weak, non-specific information, while “processibility” is capacities of supporting M. Hegarty, B. Meyer, and N. Hari Narayanan (Eds.): Diagrams 2002, LNAI 2317, pp. 116–130, 2002. c © Springer-Verlag Berlin Heidelberg 2002 The Inferential-Expressive Trade-Off: A Case Study of Tabular Representations 117 Fig. 1. A well-formed representation of the system Rt of feature tables. efficient inferences on the user’s part. Stenning and Inder [7] even propose a general ordering of representation systems according to their expressive flexibility, and compare the inferential potentials of systems in different places in this ordering. Thus, previous studies strongly suggest an association among (a) the potential of a graphical system to allow efficient inferences, (b) its capacity for rich presentation of information, and (c) its tendency to prohibit flexible presentations of information. What is then the semantic mechanism behind this association? Is there any common property of graphical representation systems from which these positive and negative traits all derive? Let us call this question the trade-off problem. In this paper, we try to answer this question by investigating a particular system of graphical representations in detail. It is a system of tabular representations of a common type, which, despite its simple syntax and semantics, exhibit the apparent association in question. Relying on previous studies of inferential and expressive potentials of graphics, we will start with showing the exact senses in which this tabular system supports efficient inferences, enables rich expression of information, and prohibits flexible expression of information (section 1). After introducing basic semantic concepts, we will show that the tabular system has a property that can be called “trackings of capturing relations over homomorphic exhaustive sets” (section 2). This property is a near-sufficient condition for all the relevant functional traits, and hence can be considered the semantic mechanism behind the trade-off phenomenon. We will close the paper by discussing the extent and generalizability of our analysis (section 3). Discussions will be kept informal throughout this paper, for compact exposition of core ideas. 1 Reproducing the Trade-Off The tabular system we investigate consists of simple “feature” tables such as the one in Figure 1. Here, the labels for rows, “A,” “B,” ... and “G,” are the names of ink-jet printer models and the labels for columns, “F1,” “F2,” ... and “F6,” are the names of various functions that ink-jet printers may have. (For the ease of reference, we use these simple names for printer models and their potential functions, instead of real names and descriptions, such as “Epson PM750,” “Canon M70,” “Print on A3-size paper” and “Print in 720× 360 dpi.”) The semantics of this system is natural one: if a circle appears in the intersection of the raw labeled by “X” and the column labeled by “Y ,” then it indicates that the